Here the information for canton Zurich.
https://ma.zh.ch/internet/sicherheit...B6rigen+IW.pdf
You will need:
- Work contract
- Certificate that you did not request social aid within the last 3 years. (Unemployment benefit is not social aid)
- Debt register extract over the last 3 years.
With permit C it is more a bureaucratic hurdle as you have the right to change canton. Never the less you need approval from the new canton before you move or you could be fined.
I couldn't not finde more information about the process in canton Zurich. Contact the Migrationsamt and ask them directly what you should do:
https://ma.zh.ch/internet/sicherheit...ufenthalt.html